Question

Sin x + cos x = -корінь з 2

Answer 1

sinxcosП/4+sinП/4cosx=-1

sin(x+П/4)=-1

x+П/4=3/2П+2Пk

x=5П/4+2Пk

Answer 2

To solve the equation sin(x) + cos(x) = -√2, we can use a few trigonometric identities.

First, we can square both sides of the equation:

(sin(x) + cos(x))^2 = (-√2)^2 sin^2(x) + 2sin(x)cos(x) + cos^2(x) = 2

Next, we can use the identity sin^2(x) + cos^2(x) = 1 to simplify the left-hand side:

1 + 2sin(x)cos(x) = 2 2sin(x)cos(x) = 1

Now, we can use the identity sin(2x) = 2sin(x)cos(x) to get:

sin(2x) = 1/2

From here, we can use the inverse sine function to solve for x:

2x = arcsin(1/2) + k(2π) or 2x = π - arcsin(1/2) + k(2π)

where k is any integer.

Simplifying each expression, we get:

x = π/4 + k(π) or x = 3π/4 + k(π)

where k is any integer.

Therefore, the solutions to the equation sin(x) + cos(x) = -√2 are:

x = π/4 + k(π) or x = 3π/4 + k(π), where k is any integer.